Method for increasing accuracy of measurement of mean polarization mode dispersion

ABSTRACT

A method of determining a mean square differential group delay associated with a length of optical fiber. The method including measuring a polarization mode dispersion vector as a function of frequency, using a frequency-domain polarization mode dispersion measurement apparatus. The method also including calculating a second-order polarization mode dispersion vector as a function of frequency by calculating a derivative of the polarization mode dispersion vector with respect to frequency. Also, calculating the mean of the magnitude of the second-order polarization mode dispersion vector to obtain a first result. Further, multiplying a proportionality coefficient by the first result to calculate the mean square differential group delay.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims priority to parent patent application Ser. No.10/747,804 filed Dec. 29, 2003. Continuity is claimed since thisapplication is a divisional of U.S. patent application Ser. No.12/702,968 filed Feb. 9, 2010, which in turn was a divisionalapplication of and claims priority to related U.S. patent applicationSer. No. 11/860,593 filed Sep. 25, 2007, which in turn was a divisionalapplication of and claims priority to U.S. patent application Ser. No.10/747,804 filed Dec. 29, 2003 and issued as U.S. Pat. No. 7,292,322 onNov. 6, 2007, which are all incorporated herein by reference.

FIELD OF THE INVENTION

The invention relates to the field of high-speed opticaltelecommunication systems, and more particularly to a method forincreasing the accuracy of the measurement of mean polarization modedispersion in optical fiber.

BACKGROUND OF THE INVENTION

The optical waveguides or fibers used to transmit signals in opticaltelecommunication systems are characterized, in part, by the vectorproperty of polarization mode dispersion (PMD). Polarization modedispersion occurs as a result of birefringence in the fiber, which maybe caused by physical asymmetry in the fiber construction itself, or bystress, strain, or other external forces imposed on the fiber. Inaddition, random polarization coupling can occur, giving rise to atime-varying statistical factor. Optical fibers display an anisotropy inthe refractive index, which will vary as a function of position andtime. Consequently, components of an optical signal that differ inpolarization will propagate at different velocities, resulting in adifferential group delay (DGD) between the components, and causingsignificant broadening of the optical pulses propagating along longlengths of fiber.

The PMD is fully characterized by a vector quantity {right arrow over(τ)}(ω) where the DGD is the magnitude of the vector |{right arrow over(τ)}(ω)|. As shown in FIG. 1, the DGD is generally designated as τ 10.Any state of polarization (SOP) can be resolved into directionalcomponents along two orthogonal principal states of polarization (PSP)12, 14. The DGD or τ 10 then represents the separation in time betweenfast PSP 12 and slow PSP 14, after traversing a length of optical fiber16. For each optical frequency or wavelength propagating in a fiber,there always exists two PSPs, such that the pulse spreading due to thefirst-order PMD vanishes if only one PSP is excited. The PMD istypically characterized in terms of an average DGD corresponding todifferent frequencies, and is independent, to first order, ofwavelength, temperature, and external perturbations. In low mode coupledfiber, this measure of DGD averaged over a large range of opticalfrequencies is fairly constant over time, but in high mode coupledfibers, for example, in long fiber spans, the frequency-averaged DGDvaries randomly in time, due to the combined effects of the variationsin birefringence and random polarization mode coupling along the fiberlength. This statistical variation in DGD lends itself tocharacterization of the DGD in terms of a statistical figure of merit,mean DGD.

Higher orders of the polarization mode dispersion also exhibitstatistical properties. The effect of second-order polarization modedispersion (SOPMD) 18 is shown in FIG. 1. The SOPMD is the firstderivative of the PMD with respect to frequency, representing the changein the PMD as a function of frequency. The SOPMD, therefore,additionally characterizes the overall pulse spreading due to thefrequency-dependence of the PMD and the spectral bandwidth of theinjected optical pulse 19.

The polarization mode dispersion of a fiber is unlike most other sourcesof degradation in an optical telecommunication system, in its dependenceon both time and frequency. Conventional methods for characterizing thefull PMD vector over a frequency range, well known by those skilled inthe art, include the Poincaré Sphere Analysis (PSA), the Jones MatrixEigenvalue (JME), Müller Matrix Method (MMM), Fixed Analyzer andinterferometric techniques. These methods provide a measure of mean DGDand root mean square (RMS) DGD, which is calculated from the set offrequency-dependent DGD values. It is then commonly assumed by thoseskilled in the art that the statistical DGD follows a Maxwelliandistribution, so that a true mean DGD

τ

, determined by averaging the DGD values obtained for a number of fibersover a bandwidth B of optical frequencies, can be estimated bymultiplying the measured RMS DGD

$\sqrt{\left\langle \tau^{2} \right\rangle_{B}}$by a factor of

$\sqrt{\frac{8}{3\pi}}.$

The fundamental problem in accurately evaluating a statisticallimitation to an estimation of the mean DGD of a fiber, in order to finda more precise measurement of the mean DGD, was first recognized in apaper by N. Gisin, B. Gisin, J. P. Von der Weid, and R. Passy, entitled“How Accurately Can One Measure a Statistical Quantity LikePolarization-Mode Dispersion?” IEEE Photon. Tech. Lett., Vol. 12, pp.1671-1673 (August 1996), which is incorporated herein by reference. Theaccuracy of mean DGD estimation does improve as the mean is taken over alarger spectral bandwidth (approaching the ideal theoretical case whereB→∞). However, contrary to the statistical requirement that each of themeasurements used to calculate an average be independent, the DGD atnearby wavelengths are not frequency independent. Gisin et al.demonstrated that this frequency dependence resulted in loweruncertainty in the mean DGD (around 9%) for larger PMD on the order of 1picosecond (ps) e.g., as compared to a 28% uncertainty in mean DGDmeasurement when the PMD is smaller (on the order of 0.1 ps). Theuncertainty in mean DGD measurement increases with decreasing sourcebandwidth. Gisin et al. demonstrated that the same level of uncertaintyis intrinsic to all measurement techniques that average the DGD overwavelength.

The mathematical formalism was developed further by M. Shtaif and A.Mecozzi, “Study of the Frequency Autocorrelation of the DifferentialGroup Delay in Fibers with Polarization Mode Dispersion,” IEEE Photon.Tech. Lett., Vol. 25, pp. 707-709 (May 2000), which is incorporatedherein by reference. In measurements of the frequency autocorrelation ofthe DGD, the square DGD, and orientation of the PMD vector, Shtaif etal. showed that all corresponding correlation bandwidths are comparable.Shtaif et al. also showed that all statistical properties of the PMDcharacterizing the fiber under test are uniquely defined by the meanDGD.

Polarization mode dispersion (PMD) is recognized as a potentiallylimiting impairment for high-speed long-haul optical transmission.Moreover, precise measurement of the true mean differential group delay(DGD) of individual fiber links and whole fiber routes is important foraccurate estimation of service outage probabilities. Since PMD varieswith time, as well as with frequency, measurements of the meanfrequency-averaged DGD of the same fiber taken at different times maydiffer from each other and from the true value of mean DGD for a givenfiber. For DGD values in the usual range of interest, and within theoptical bandwidths of commercially available equipment, the variance ofDGD measurements is approximately inversely proportional to the opticalbandwidth of the optical source used for the measurement. In otherwords, an accurate measurement of the mean DGD of low birefringencefiber is limited by the optical bandwidth of the source used for themeasurement.

The need for precise PMD characterization will increase as thehigh-speed networks of the future employ very low PMD fibers. Thereexists a need, therefore, for more precise measurement of the mean DGDof individual fiber links and whole fiber routes than is presentlyprovided by conventional methods.

SUMMARY OF THE INVENTION

The present invention, which addresses the needs unmet by conventionalmethods, relates to methods of improving accuracy of measuring adifferential group delay (DGD) in an optical fiber link and in a wholeoptical fiber route.

A method of the present invention for measuring a true mean differentialgroup delay

τ

of at least a length of optical fiber includes the initial step ofmeasuring a mean square differential group delay

τ²

_(B) averaged over a finite spectral bandwidth B of a source, using apolarization mode dispersion measurement apparatus. A root mean squaredifferential group delay is then calculated in accordance with

$\sqrt{\left\langle \tau^{2} \right\rangle_{B}}$and a systematic correction factor ε to the conventional method ofestimating true mean

τ

from the measured root mean square differential group delay

$\sqrt{\left\langle \tau^{2} \right\rangle_{B}}$is applied. The systematic correction factor ε minimizes a systematicerror caused by the finite spectral bandwidth of the source.

Preferably, the systematic correction factors is applied to the meansquare differential group delay in accordance with

${\left\langle \tau \right\rangle = {\sqrt{\frac{8}{3\pi}\left\langle \tau^{2} \right\rangle_{B}} + \varepsilon}},$to obtain the true mean differential group delay

τ

. In the regime where τ_(RMS)B>>1,

τ

is calculated according to:

$\begin{matrix}{{\left\langle \tau \right\rangle = {\sqrt{\frac{8}{3\pi}\left\langle \tau^{2} \right\rangle_{B}} + {\frac{8}{9\sqrt{2}}\frac{1}{B}}}},} & {\left( {16a} \right)\mspace{14mu}{and}\mspace{14mu}\left( {16b} \right)}\end{matrix}$in other words, ε is

$\frac{8}{9\sqrt{2}}{\frac{1}{B}.}$

This method, which is applied directly to a measured mean squaredifferential group delay, may be applied to measurements taken usingtime-domain techniques with an apparatus such as an interferometer. Thismethod may also be applied to measurements taken using frequency-domaintechniques, such as Jones Matrix Eigenanalysis, Poincaré SphereAnalysis, and Müller Matrix Method using an apparatus, for example,including a polarimeter. The method may be used for measuring the meandifferential group delay through a single optical fiber link, or anentire optical fiber route.

In another embodiment of the method of the present invention, a methodfor measuring a mean differential group delay

τ

of at least one length of optical fiber, includes an initial step ofcharacterizing a polarization mode dispersion vector as a function offrequency using a frequency-domain polarization mode dispersionmeasurement apparatus. The method further includes calculating asecond-order polarization mode dispersion vector as a function offrequency {right arrow over (τ)}_(ω) from the polarization modedispersion vector, and calculating a mean of the square root of amagnitude of the second-order polarization mode dispersion vector|{right arrow over (τ)}_(ω)| to obtain a first result, according to

|{right arrow over (τ)}_(ω)|^(1/2)

. The first result is multiplied by a proportionality coefficient A₂ tocalculate the mean differential group delay, in accordance with thefollowing equation:A ₂

|{right arrow over (τ)}_(ω)|^(1/2)

=

τ

.   (21)

Preferably, the proportionality coefficient A₂ is obtained from theprobability density function of the second-order polarization modedispersion vector. Most preferably, A₂ is substantially equal to 1.37.

A further embodiment of the method of the present invention provides amethod for measuring a mean differential group delay

τ

of a length of optical fiber, including an initial step of measuring amagnitude of a polarization mode dispersion vector as a function offrequency, using a frequency-domain polarization mode dispersionmeasurement apparatus, where the magnitude of the polarization modedispersion vector is a scalar differential group delay. The methodfurther includes calculating a frequency-derivative of the scalardifferential group delay, the frequency derivative being a scalarsecond-order polarization mode dispersion function. The method furtherincludes calculating a first result, according to

$\left\langle {\frac{\mathbb{d}{\overset{\rightarrow}{\tau}}}{\mathbb{d}\omega}}^{\frac{1}{2}} \right\rangle,$and multiplying a proportionality coefficient B₂ by the first result.The mean differential group delay is calculated, therefore, inaccordance with the following equation:

$\begin{matrix}{{B_{2}\left\langle {\frac{\mathbb{d}{\overset{\rightarrow}{\tau}}}{\mathbb{d}\omega}}^{\frac{1}{2}} \right\rangle} = {\left\langle \tau \right\rangle.}} & (26)\end{matrix}$

Preferably, B₂ is obtained from the probability density function of thescalar second-order polarization mode dispersion function. Mostpreferably, B₂ is substantially equal to 2.64.

Yet another embodiment of the present invention provides a method formeasuring a mean square differential group delay τ² _(RMS) of a lengthof optical fiber, including an initial step of characterizing apolarization mode dispersion vector as a function of frequency using afrequency-domain polarization mode dispersion measurement apparatus.Additionally, a second-order polarization mode dispersion vector iscalculated as a function of frequency {right arrow over (τ)}_(ω) fromthe polarization mode dispersion vector. The method further includescalculating a mean of the magnitude of the second-order polarizationmode dispersion vector |{right arrow over (τ)}_(ω)| to obtain a firstresult, according to

|{right arrow over (τ)}_(ω)|

. The first result is multiplied by a proportionality coefficient A₁ tocalculate the mean square differential group delay, in accordance withthe following equation:A₁

|{right arrow over (τ)}_(ω)|

=τ² _(RMS).   (20)

Preferably, A₁ is obtained from the probability density function of thesecond-order polarization mode dispersion vector. Most preferably, A₁ issubstantially equal to 2.02.

Still another embodiment of the present invention provides a method formeasuring a mean square differential group delay τ² _(RMS) of a lengthof optical fiber, including an initial step of measuring a magnitude ofa polarization mode dispersion vector as a function of frequency, usinga frequency-domain polarization mode dispersion measurement apparatus,where the magnitude of the polarization mode dispersion vector is ascalar differential group delay. The method further includes calculatinga scalar second-order polarization mode dispersion function, the scalarsecond-order polarization mode dispersion function being afrequency-derivative of the scalar differential group delay.Additionally, a first result is calculated according to

$\left\langle {\frac{\mathbb{d}{\overset{\rightarrow}{\tau}}}{\mathbb{d}\omega}} \right\rangle.$The first result is multiplied by a proportionality coefficient, B₁, tocalculate the mean square differential group delay, in accordance withthe following equation:

$\begin{matrix}{{B_{1}\left\langle {\frac{\mathbb{d}{\overset{\rightarrow}{\tau}}}{\mathbb{d}\omega}} \right\rangle} = {\tau_{RMS}^{2}.}} & (25)\end{matrix}$

B₁ is preferably obtained from the probability density function of thescalar second-order polarization mode dispersion function. B₁ is mostpreferably substantially equal to 6.80.

The measurement of polarization mode dispersion through an optical fibermay be through a single optical fiber link or a whole optical fiberroute in an optical telecommunication network.

In an additional embodiment of the method of the present invention, amethod is provided for characterizing polarization mode dispersion of alength of optical fiber using a source of bandwidth B. The methodincludes an initial step of collecting polarization mode dispersion dataas a function of frequency from a frequency-domain polarization modedispersion measurement apparatus. The method further includes extractingone of a vector and a scalar frequency-dependent function from thepolarization mode dispersion data, by applying a frequency-domainpolarization mode dispersion technique, the one of the vector and thescalar function being one of a first-order and second-order polarizationmode dispersion function. A systematic correction is applied thereto,wherein the systematic correction minimizes a systematic error inconventional methods of measuring mean differential group delay, causedby the finite source bandwidth B. Applying the systematic correctionresults in a derivation of one of a mean differential group delay

τ

and a mean square DGD τ² _(RMS).

Another method of measuring a mean differential group delay

τ

of a length of optical fiber according to the present invention includesthe steps of deriving a first mean

τ

in accordance with equation (21), and a second mean

τ

in accordance with equation (26), and then deriving a linear equation ofa sum of the first mean

τ

and the second mean

τ

to calculate a combined mean

τ

. A sum of coefficients of the linear equation is substantially equal toone.

Yet another method of measuring a mean square differential group delayτ² _(RMS) of a length of optical fiber in accordance with the presentinvention includes the steps of deriving a first mean squaredifferential group delay τ² _(RMS) in accordance with equation (20),deriving a second mean square differential group delay τ² _(RMS) inaccordance with equation (25), and deriving a linear equation of a sumof the first mean square differential group delay τ² _(RMS) and thesecond mean square differential group delay τ² _(RMS) to calculate acombined mean square differential group delay τ² _(RMS). A sum ofcoefficients of the linear equation is substantially equal to one.

Other objects and features of the present invention will become apparentfrom the following detailed description considered in conjunction withthe accompanying drawings. It is to be understood, however, that thedrawings are designed as an illustration only and not as a definition ofthe limits of the invention.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a pictorial view of an optical pulse propagating a length ofbirefringent optical fiber.

FIG. 2 is a flow diagram illustrating a prior art method for calculatingmean differential group delay (DGD).

FIG. 3 is a block diagram illustrating a conventional interferometricpolarization mode dispersion (PMD) measuring device.

FIG. 4 is a block diagram illustrating a conventional apparatus used ina frequency-domain PMD measuring device.

FIG. 5 is a plot of the standard deviation of the distribution ofsimulated mean DGD measurements of

$\left\langle \tau \right\rangle_{B},{A_{2}\left\langle {{\overset{\rightarrow}{\tau}}_{\omega}}^{\frac{1}{2}} \right\rangle_{B}},{{and}\mspace{14mu} B_{2}{\left\langle {\frac{\mathbb{d}{\overset{\rightarrow}{\tau}}}{\mathbb{d}\omega}}^{\frac{1}{2}} \right\rangle_{B}.}}$

FIGS. 6 a and 6 b are plots showing the improved standard deviationobtained with a systematic correction applied according to the method ofthe present invention.

FIG. 7 is a plot of the standard deviation of the distribution ofsimulated square mean DGD measurements of

$\left\langle \tau^{2} \right\rangle_{B},{A_{1}\left\langle {{\overset{\rightarrow}{\tau}}_{\omega}} \right\rangle_{B}},{{and}\mspace{14mu} B_{1}{\left\langle {\frac{\mathbb{d}{\overset{\rightarrow}{\tau}}}{\mathbb{d}\omega}} \right\rangle_{B}.}}$

FIG. 8 is a plot of standard deviation resulting from combining themethods shown in FIG. 5.

DETAILED DESCRIPTION

The present invention relates to improving the accuracy of calculating atrue mean differential group delay (DGD) or a true mean-square DGD

τ

from a measurement of polarization mode dispersion (PMD). The presentinvention provides a method for correcting a systematic error in aconventional conversion of measured mean-square DGD measurements to atrue mean DGD

τ

. The systematic error stems from the use of a measurement source of afinite bandwidth B, and from the conventional conversion applied tomeasured root mean square DGD

$\sqrt{\left\langle \tau^{2} \right\rangle_{B}}$to obtain an estimate of true mean DGD

τ

. By incorporating improved averaging techniques to the measured rootmean square DGD

$\sqrt{\left\langle \tau^{2} \right\rangle_{B}}$obtained with conventional PMD measurement apparatus, the presentinvention offers improved resolution in true mean

τ

or true mean-square DGD

τ²

measurements.

The present invention also provides a method to improve accuracy inmeasurements of the true mean DGD

τ

of optical fiber using statistics of a second-order polarization modedispersion (SOPMD) and incorporating improved averaging techniques. A30% reduction in measurement uncertainty is obtained when using themethod in accordance with the present invention, which is equivalent todoubling the bandwidth B of an optical source.

The following notation and definitions are used herein. Opticalfrequency ω refers to 2πc/λ, where λ is the wavelength of the source andc is the speed of light. Optical wavelength and frequency are usedinterchangeably to characterize the measurement source. Analogously,spectral bandwidth and frequency bandwidth both refer to the opticalsource characteristics in wavelength and frequency space, respectively.

An average over a finite angular frequency bandwidth B of a variableX(ω) associated with a given fiber is denoted and defined as follows:

$\begin{matrix}{\left\langle X \right\rangle_{B} = {\frac{1}{B}{\int_{{- B}/2}^{B/2}{{X(\omega)}{{\mathbb{d}\omega}.}}}}} & (1)\end{matrix}$A “measurement of the mean” refers to a finite bandwidth average, asdescribed by equation (1), applied to an experimentally measurablequantity.

Applying equation (1), a root mean square of a parameter X is thendefined as:

$\begin{matrix}{{\sqrt{\left\langle X^{2} \right\rangle}}_{B} = {\sqrt{\frac{1}{B}{\int_{{- B}/2}^{B/2}{{X^{2}(\omega)}{\mathbb{d}\omega}}}}.}} & (2)\end{matrix}$A root mean square (RMS) of the DGD measured as a function of frequencyover a source of and width B is therefore denoted generally by:

$\begin{matrix}{{{{Measured}\mspace{14mu}{RMS}\mspace{14mu}{DGD}} = \sqrt{\left\langle \tau^{2} \right\rangle_{B}}},} & (3)\end{matrix}$where subscript “B” indicates that measurements are taken over a finitesource bandwidth, B.On the other hand a “true” RMS DGD, obtained by measuring over infinitebandwidth in equation (2) is denoted herein as τ_(RMS).

Equation (3) may be written as an integral of a function τ²(ω) inaccordance with equation (2). As well understood by those skilled in theart, however, when measuring polarization mode dispersion in thefrequency domain, a DGD

τ

is measured as the time delay between the principle states ofpolarization (PSP) associated with a discrete frequency, orequivalently, as the magnitude of the PMD measured at each discretefrequency over the source bandwidth B. An RMS DGD value over a finitebandwidth B may therefore be calculated as an average of the discretefrequency-dependent DGD values over the source bandwidth, in accordancewith the following:

$\begin{matrix}{{\sqrt{\left\langle \tau^{2} \right\rangle_{B}} = \sqrt{\frac{1}{B}{\sum\limits_{i = {- N}}^{N}{{\tau^{2}({\mathbb{i}\Delta\omega})}{\Delta\omega}}}}},} & (4)\end{matrix}$where Δω=B/(2N+1), and a total of 2N+1 measurements are made over sourcebandwidth B.

A mean over an ensemble of N fibers is denoted by a bar over theparameter averaged, such as X in the following equation:

$\begin{matrix}{\overset{\_}{X} = {\frac{1}{N}{\sum\limits_{k = 1}^{N}{X_{k}.}}}} & {(5).}\end{matrix}$Averaging of the same frequency-dependent parameter, the DGD forexample, over an infinite number of fibers (N→∞) with identicallydistributed birefringence, where the parameter is measured for eachfiber over a finite bandwidth B, is equivalent to measuring the samefrequency-dependent parameter of one fiber over an infinite bandwidth(B→∞). For this “real” or “true” mean, taken over the limit of infinitebandwidth, the subscript “B” is omitted.

A second-order polarization mode dispersion (SOPMD), which is thederivative of the polarization mode dispersion vector with respect tofrequency, is represented by τ_(ω).

A fiber link may refer to any fiber optic cable or cable assemblyconnecting two points, such as optical amplifiers, along an opticaltransmission line. A fiber route may include multiple fiber links, andincludes the total fiber optic cable between any two terminating pointsin an optical transmission line.

In a first embodiment, a method is provided to more accurately measurethe mean of the differential group delay from conventional PMDmeasurement techniques. The method employs a systematic method tominimize a systematic error that occurs when converting root-mean-squaremeasurements of the spectral-averaged DGD to the true mean DGD

τ

using conventional means. The error is a consequence of the finitebandwidth B of the measurement source.

Referring to FIG. 2, a conventional method 20 of estimating the truemean DGD

τ

includes the step 22 of characterizing the PMD vector using techniquescommonly known, such as Jones Matrix Eigenanalysis (JME), Müller MatrixMethod (MMM), and Poincaré Sphere Analysis (PSA). In step 23, the meanDGD is calculated as an average over the spectral source bandwidth toestimate a true mean. Alternatively, in step 24, the RMS value of theDGD

$\sqrt{\left\langle \tau^{2} \right\rangle_{B}}$is calculated from the magnitude of the frequency-averaged PMD vector,in accordance with equation (4), for example. Alternately, step 26 isused to derive

$\sqrt{\left\langle \tau^{2} \right\rangle_{B}}$directly from an interferometric measurement of the square of time delayaveraged over the source bandwidth. In step 28,

$\sqrt{\left\langle \tau^{2} \right\rangle_{B}}$derived from either step 24 or step 26 is multiplied by a conventionalcorrection factor to calculate an estimate of true mean

τ

29.

It has been commonly assumed in the prior art that the DGD can bestatistically represented by a Maxwellian distribution. Therefore, basedon statistical probability theory, an estimate of mean DGD is calculatedby measuring

$\sqrt{\left\langle \tau^{2} \right\rangle_{B}}$for a fiber and estimating the mean that would be found for an ensemblecomprised of an infinite number of fibers in accordance with equation(5), by multiplying the result by the conventional correction factor of

$\sqrt{\frac{8}{3\pi}}.$In other words, by conventional methods, the following equation isapplied to obtain an estimate of true mean

τ

29 in accordance with step 28 of FIG. 2:

$\begin{matrix}{\left\langle \tau \right\rangle \approx {\sqrt{\frac{8}{3\pi}}{\sqrt{\left\langle \tau^{2} \right\rangle_{B}}.}}} & (6)\end{matrix}$The validity of this approximation may be verified by comparing:

$\begin{matrix}{{\overset{\_}{\left\langle \tau \right\rangle_{B}}\overset{?}{->}{\sqrt{\frac{8}{3\pi}} \cdot \overset{\_}{\sqrt{\left\langle \tau^{2} \right\rangle_{B}}}}},} & (7)\end{matrix}$where the ensemble averaging is taken over a multiplicity of differentfibers with identically distributed birefringence.

Equation (7) is only correct, however, as B→∞, and introduces asystematic error when the measurement bandwidth B is finite. This is theeasiest to see in the extreme case of a very small bandwidth, that is asB→0. This case corresponds to a situation where only one frequency pointper fiber is used to make a measurement of mean DGD. Averaged over manyfibers with identically distributed birefringence, the left hand sideLHS of equation (7) will converge to

τ

, while its right hand side RHS will converge to

${\sqrt{\frac{8}{3\pi}}\left\langle \tau \right\rangle},$resulting in about an 8% systematic error.

In the method of the present invention, a function F is derived, whichis defined on the frequency span from −B/2 to B/2 to obtain an estimateof the DGD as a function of source bandwidth B, denoted as τ(B), suchthatF[τ(ω)]=τ(B).   (8)The function is defined so that for an ensemble of many different fiberswith the same τ_(RMS), its ensemble average converges to the true mean

τ

, asτ(B)=

τ

,   (9)and the standard deviation of τ(B), denoted as std(τ(B)), is minimizedrelative to the standard averaging.

To derive a systematic correction factor, the systematic error isestimated asymptotically when the product of the bandwidth B and τ_(RMS)is much greater than 1 (τ_(RMS)B>>1) and the systematic error is small.The result of any measurement of the mean square DGD averaged over afinite source bandwidth B, or

τ²

_(B), is represented by the following equation:

$\begin{matrix}{{{\frac{8}{3\pi}\left\langle \tau^{2} \right\rangle_{B}} = {\left\langle \tau \right\rangle^{2} + \eta}},} & (10)\end{matrix}$

where η represents the deviation of the measurement from the mean foundfor the ensemble, and the average of η values for the ensemble is zero (η=0). Assuming that this deviation is small, a binomial expansion can beapplied to approximate η so that equation (10) can be written in thefollowing form:

$\begin{matrix}{\sqrt{\frac{8}{3\pi}\left\langle \tau^{2} \right\rangle_{B}} \cong {\left\langle \tau \right\rangle + \frac{\eta}{2\left\langle \tau \right\rangle} - {\frac{\eta^{2}}{8\left\langle \tau \right\rangle^{3}}.}}} & (11)\end{matrix}$

Since η=0, taking an ensemble average of each side of equation (11)leaves only the η² term and

τ

on the right hand side, so that the systematic error ε in measuring themean DGD can be estimated from the following equation:

$\begin{matrix}{ɛ = {\left\langle \tau \right\rangle - \overset{\_}{\sqrt{\frac{8}{3\pi}\left\langle \tau^{2} \right\rangle_{B}}} - {\frac{\overset{\_}{\eta^{2}}}{8\left\langle \tau \right\rangle^{3}}.}}} & (12)\end{matrix}$

The systematic error ε is equivalently a systematic correction factor εthat is applied to the measured RMS DGD

$\sqrt{\left\langle \tau^{2} \right\rangle_{B}}$to improve the resolution of true mean DGD

τ

calculations.

The variance of

τ²

_(B) was evaluated in M. Shtaif and A. Mecozzi, “Study of the FrequencyAutocorrelation of the Differential Group Delay in Fibers withPolarization Mode Dispersion,” IEEE Photon. Tech. Lett., Vol. 25, pp.707-709 (May 2000) for τ_(RMS)B>>1 as follows:var

τ²

_(B)=(π²/√{square root over (2)})

τ

³ /B   (13)where the variance is defined as follows:

$\begin{matrix}{{{var}\left\langle \tau^{2} \right\rangle_{B}} = {\overset{\_}{\left( {\left\langle \tau^{2} \right\rangle_{B} - \left\langle \tau^{2} \right\rangle} \right)}}^{2}} & (14)\end{matrix}$

With the help of equation (10) to derive an expression for η² , andsubstituting the expression for the variance from equation (13), a valuefor the systematic error can be evaluated from equation (12) as follows:

$\begin{matrix}{ɛ = {{\left( \frac{8}{3\pi} \right)^{2}\frac{\overset{\_}{\left( {\left\langle \tau^{2} \right\rangle_{B} - \left\langle \tau^{2} \right\rangle} \right)^{2}}}{8\left\langle \tau^{3} \right\rangle}} = {\frac{8}{9\sqrt{2}}{\frac{1}{B}.}}}} & (15)\end{matrix}$

In order to obtain an unbiased estimate of the mean DGD

τ

, therefore, the following equation is preferably applied to themean-square DGD

τ²

_(B) measured using any of the conventional measurement techniques forobtaining the mean-square DGD:

$\begin{matrix}{{\left\langle \tau \right\rangle = {\sqrt{\frac{8}{3\pi}\left\langle \tau^{2} \right\rangle_{B}} + ɛ}},} & \left( {16a} \right) \\{{{where}\mspace{14mu} ɛ} = {\frac{8}{9\sqrt{2}}{\frac{1}{B}.}}} & \left( {16b} \right)\end{matrix}$

One embodiment of the method of the present invention, therefore,includes the steps of deriving a frequency-averaged RMS measurement ofthe DGD

τ²

_(B) from conventional PMD measuring apparatus, and applying asystematic correction to the measured RMS to obtain true mean DGD

τ

. As shown in equation (16a), the systematic correction is made byadding a systematic correction factors to the conventional approximationof

τ

that is described by equation (6). The preferred systematic correctionfactors is characterized by equation (16b) above. The measuringapparatus can include the time-domain and frequency-domain measurementdevices, as described below.

Examples of conventional measurement techniques to which this systematiccorrection can be applied include, but are not limited to,frequency-domain measurement techniques such as JME, MME, PSA, andtime-domain techniques such as low-coherence interferometry. Afundamentally equivalent method to a scanning Michelson interferometer,that also works in the frequency domain, is the Fixed AnalyzerTechnique.

A block diagram of an interferometer 30 for PMD measurements in thetime-domain is shown in FIG. 3. Time-domain techniques provide a directmeasure in the time-domain of the frequency-averaged DGD. The commonlyknown interferometric technique for measuring PMD is described in moredetail in N. Gisin, J. P. Von der Weid and J. P. Pellaux, “PolarizationMode Dispersion of Short and Long Single-Mode Fibers,” J. Lightw.Technol., Vol. 9, p. 821 (1991), which is incorporated herein byreference. In the Michelson-type interferometer 30 of FIG. 3, forexample, a spectrally broad (low-coherence) source 32 is used to couplelight 34 into the device under test (fiber) 36. The output of fiber 36is collimated using collimating optics 38 and directed onto abeamsplitter 40. The beamsplitter 40 divides the light into two pathsalong two arms of the interferometer 30. One path is directed toward amoveable mirror 42, which can be translated along a path coincident withthe direction of the incident light. The second is directed toward astationary mirror 44. When the mirror 42 is translated, interferencefringes are seen by a detector 46 as long as the time-delay differencebetween the two arms matches a delay generated in the device under test36 to within a coherence time of the source 32.

In the simplest case of a non-mode-coupled device, the interferogramprovided by a conventional interferometer 30 is symmetric andrepresented by two side lobes and a central autocorrelation peak. Thedistance from a side lobe to the central peak represents the DGDaveraged over the source spectrum or bandwidth. In the mode-coupled caseof a fiber, the interferogram includes multiple peaks representingvarious principle states of polarization (PSP) associated with differentfrequencies. In this case, as well-known to those skilled in the art,the DGD is characterized as a RMS value over the bandwidth, that is

$\sqrt{\left\langle \tau^{2} \right\rangle_{B}},$as described by equation (3). Discrete measurements of the PMD as afunction of frequency are not needed since the time-domain measurementsare inherently frequency-averaged over the source bandwidth.

FIG. 4 shows a block diagram of a common polarimeter 50 that can be usedin a measurement device for any of the frequency-based techniques, suchas PSA, JME, and MMM. In FIG. 4, a polarization state of a tunable laser52 is controlled by a polarization controller 54. The polarized outputfrom the controller 54 is coupled to the fiber under test 56. Thepolarization state of the optical signal after traversing the length offiber 56 is then analyzed with a conventional polarization stateanalyzer 58. By stepping the laser 52 in specified frequency intervalsover the entire bandwidth, an output polarization state for each inputpolarization state is obtained as a function of frequency. Eachtechnique differs in which input polarization states are used and in howthe data set of measured output polarization states are manipulated. Theresult, however, is the same: a measurement of the PMD vector, that is,both DGD and the PSP as a function of frequency.

The PSA and JME techniques are well-known to those skilled in the artand are described in detail, for example, in Normand Cyr, Andre Girard,and Gregory W. Schinn, “Stokes Parameter Analysis Method, theConsolidated Test Method for PMD Measurements”, NFOEC '99 Convention,Chicago, Ill. (1999), which is incorporated herein by reference. Thesetechniques are based on the measurement of so-called polarimetric Stokesparameters. These two techniques differ in the algorithms used todescribe the PMD from the measured Stokes parameters, as well as in therequired form of the generally three measured Stokes parameters. The JMEalgorithm requires only linear input states of polarization (SOP),whereas the PSA algorithm usually requires one of the three input statesto be circular.

The PSA and JME methods are alike, however, in requiring the same testprocedures. Both find the PSP and the DGD as a function of frequencyfrom raw polarimetric data. The raw polarimetric data is the normalizedStokes vector representing the SOP at the output of a fiber as afunction of optical frequency for a number of input SOPs. The normalizedStokes vector, ŝ, is the position of the output SOP on a Poincarésphere. Basically, both methods measure the local “angular velocity”(where time is actually optical frequency here) of the output SOP from ameasurement of its position on the sphere at two neighboring opticalfrequencies ω and (ω+Δω). One difference is that PSA is performeddirectly in the Stokes parameter representation of the SOP, while JME isperformed in the Jones vector representation, after the Stokes vectorshave been transformed into normalized Jones vectors.

The MMM technique has been described as a combination of the PSA and JMEmethod. The MMM method is described in detail in R. M. Jopson, L. E.Nelson, and H. Kogelnik, “Measurement of Second-Order Polarization-ModeDispersion Vectors in Optical Fibers”, IEEE Phot. Tech. Lett., Vol. 11,No. 9, pp. 1153-55 (September 1999). Like the other two techniques, theMMM requires the determination of transmitted output Stokes vectorscorresponding to two input polarizations s_(i) at each frequency ω.Unlike PSA however, MMM has the ability to determine the rotation matrixof the fiber. It also stays entirely in Stokes vector space. Thetechnique is advantageously independent of the relative angle betweenthe two input linear polarizations. MMM may also be used with ainterleaving method to measure the first- and second-order PMD vectorsof fibers, as described in Jopson, et al.

Both the interferometer of FIG. 3 and the polarimeter of FIG. 4, as wellas any conventional PMD apparatus, can be used to producefrequency-averaged RMS measurements of the DGD over the sourcebandwidth. The systematic correction obtained above in equation (16b)can then be directly applied to the measured RMS DGD τ_(RMS) as inequation (16a) to minimize the error in calculating the true mean DGD

τ

.

The method of the present invention provides a systematic correction forthe property that measurements of

$\left\langle \tau \right\rangle_{B}\mspace{14mu}{and}\mspace{14mu}\sqrt{\frac{8}{3\pi}\left\langle \tau^{2} \right\rangle_{B}}$are not equivalent, due to the physical limitation imposed by the finitesource bandwidth. In other words, taking the square root of afrequency-averaged parameter leads to systematic errors for finitebandwidth measurements. The systematic correction for this property canalso be extended to higher powers of DGD and to frequency derivatives ofPMD.

In another embodiment of the present invention, a method of moreaccurately measuring mean DGD

τ

is provided by measuring a second-order PMD (SOPMD) over a finitebandwidth B and applying a systematic correction to minimize the randomerrors that typically occur from the measurement over a finitebandwidth. The systematic correction includes an averaging procedurethat avoids the systematic error induced by conventional methods oftaking roots after averaging over frequency. This method producessuperior results when the fiber length over which measurements areperformed is considerably longer than the birefringence correlationlength, as is the case for fiber spans in telecommunication systems.

It is shown in G. J. Foschini and C. D. Poole, “Statistical Theory ofPolarization Dispersion in Single Mode Fibers,” IEEE Journ. of LightwaveTechn., Vol. 9, pp. 1439-1456 (November 1991) (“Foschini and Poole”),that the mean of the SOPMD in a fiber, represented by

τ_(ω)

depends only on its mean square DGD

τ²

when the fiber is much longer than the correlation length of itsbirefringence, in accordance with the following equation:

$\begin{matrix}{\left\langle \tau_{\omega}^{2} \right\rangle = {\frac{1}{3}{\left\langle \tau^{2} \right\rangle^{2}.}}} & (17)\end{matrix}$

It has also been shown that the SOPMD can be accurately measuredexperimentally if the measurements are performed with a small enoughfrequency step. (See R. M. Jopson, L. E. Nelson, H. Kogelnik,“Measurement of Second-Order Polarization-Mode Dispersion Vectors inOptical Fibers”, IEEE Phot. Tech. Lett, Vol. 11, pp. 1153-55 (September1999), incorporated herein by reference.) Jopson et al. applied aninterleaving technique to increase the frequency resolution of themeasurements, then applied the Müller Matrix Method (MMM) to thesemeasurements to obtain the PMD vector, and calculated the RMS DGDτ_(RMS) from the magnitude of the PMD vector {right arrow over (τ)}. Inaddition, Jopson et al. calculated the SOPMD from these measurements andapplied the Foschini and Poole relation described by equation (20) toestimate either the mean square DGD

τ²

or the root mean square DGD τ_(RMS).

As described above, however, due to the finite bandwidth B of the sourceused for the measurement, a systematic error occurs when taking thesquare root of a mean square value averaged over frequency. The directapplication of equation (17) according to the prior art, therefore,produces inaccurate results, especially over low source bandwidths. Theerror is present whether it is a first, second, or higher order of PMDthat is measured as a frequency-averaged RMS value. The estimate of themean DGD

τ

from second-order PMD measurements, therefore, will also be biased atsmall measurement bandwidths for reasons very similar to the embodimentdescribed above with respect to first order PMD measurements.

An additional embodiment of the method of the present invention,therefore, provides an accurate measure of the mean DGD

τ

from measurements of the SOPMD taken over a finite bandwidth B, byapplying a systematic correction to minimize the random errors thatoccur from the measurement over a finite bandwidth. The systematiccorrection includes an averaging technique that avoids the systematicerror induced by conventional methods of taking roots after averagingover frequency.

A further embodiment of the method of the present invention provides anaccurate measure of the mean squared DGD τ² _(RMS) from measurements ofthe SOPMD taken over a finite bandwidth, by applying a systematiccorrection to minimize the random errors that occur from the measurementover a finite bandwidth. The systematic correction includes an averagingtechnique that avoids the systematic error induced by conventionalmethods of taking roots after averaging over frequency.

The SOPMD is the first derivative of the PMD with respect to frequency,representing the change in the PMD as a function of frequency.Specifically, if {right arrow over (τ)}(ω) denotes the PMD vector for afrequency ω, defined by the principal states of polarization (PSP) atthat frequency ω, then the magnitude of the vector |{right arrow over(τ)}(ω)| is the differential group delay DGD τ or the set of PSP definedfor frequency ω. Therefore, a set of vector PMD data as a function offrequency ω and a set of scalar DGD data as a function of frequency ωmay be generated from any of the methods used to extract the PMD from afrequency-domain measurement device. Such measurement device maycomprise a polarimeter or a Fixed Analyzer, for example.

In addition, a second-order PMD can be defined as either a vector orscalar quantity. The vector SOPMD is denoted as {right arrow over(τ)}_(ω) and is found from the set of vector PMD data, according to theconventional method of defining a derivative from a discrete data set ofn sampled frequency points as follows:

$\begin{matrix}{{\overset{\rightarrow}{\tau_{\omega}}\left( \omega_{n} \right)} = {\frac{{\overset{\rightarrow}{\tau}\left( \omega_{n + 1} \right)} - {\overset{\rightarrow}{\tau}\left( \omega_{n} \right)}}{\omega_{n + 1} - \omega_{n}}.}} & (18)\end{matrix}$

The scalar SOPMD is more easily calculated as the derivative of thescalar DGD and is denoted as

$\frac{\mathbb{d}{{\overset{\rightarrow}{\tau}(\omega)}}}{\mathbb{d}\omega},$where again, the magnitude of the vector |{right arrow over (τ)}(ω)| isthe differential group delay DGD τ, a scalar quantity. This scalar SOPMDis found directly from the set of frequency-dependent scalar DGD τvalues, i.e. from τ(ω), according to the conventional method of defininga derivative from a discrete data set as follows:

$\begin{matrix}{{\tau_{\omega}\left( \omega_{n} \right)} = {\frac{{\tau\left( \omega_{n + 1} \right)} - {\tau\left( \omega_{n} \right)}}{\omega_{n + 1} - \omega_{n}}.}} & (19)\end{matrix}$

Different quantities scaling as τ and τ² respectively are preferablyused to avoid bias in the calculated results for each of theseembodiments. A probability density function (PDF) for the magnitude ofthe scalar SOPMD and the vector SOPMD have been calculated by Foschini,et al. in “Probability Densities of Second-Order Polarization ModeDispersion Including Polarization Dependent Chromatic Fiber Dispersion,”IEEE Phot. Tech. Lett., Vol. 12, pp. 293-295 (March 2000), (Foschini etal.), which is incorporated herein by reference.

By integrating the probability density function for the magnitude of thevector SOPMD in conjunction with equation (17), the mean square and themean of the DGD may be derived, resulting in the followingrelationships:

$\begin{matrix}{{{A_{1}\left\langle {{\overset{\rightarrow}{\tau}}_{\omega}} \right\rangle} = \tau_{RMS}^{2}},} & (20) \\{{A_{2}\left\langle {{\overset{\rightarrow}{\tau}}_{\omega}}^{\frac{1}{2}} \right\rangle} = {\left\langle \tau \right\rangle.}} & (21)\end{matrix}$Coefficients

$A_{1} = {{\frac{3\pi^{2}}{16G} \approx {2.02\mspace{14mu}{and}\mspace{14mu} A_{2}}} = {\left\lbrack {\frac{3}{2\sqrt{\pi}}{\sum\limits_{k = 0}^{\infty}\;\frac{\left( {- 1} \right)^{k}}{\left( {{2k} + 1} \right)^{\frac{3}{2}}}}} \right\rbrack^{- 1} \approx 1.37}}$(Catalan constant G≈0.916) may be calculated from the PDF of the SOPMD,and verified by simulations. Written this way, equations (20) and (21)provide a way to estimate, without bias, the mean DGD and the meansquare DGD using the SOPMD vector magnitude |{right arrow over(τ)}_(ω)|.

PMD and SOPMD measurements are strongly correlated as seen from therelation for the conditional mean square of the SOPMD, derived in J. P.Gordon, “Statistical Properties of Polarization Mode Dispersion”, Proc.VSS '02 (Venice, Italy 2002) and provided below:

$\begin{matrix}{\left\langle {\tau_{\omega}^{2}❘\tau} \right\rangle = {{\frac{2}{9}\tau^{2}\left\langle \tau^{2} \right\rangle} + {\frac{1}{9}{\left\langle \tau^{2} \right\rangle^{2}.}}}} & (22)\end{matrix}$Therefore, if a specific measurement of the mean DGD of a given fiber isabove the “true” mean, the mean DGD estimated from equation (21) will beoverestimated as well. Gordon, however, has shown that the conditionalmean square of the component of the second order PMD vector that isparallel to the direction of the PMD vector is independent of the PMDvector length and can also be expressed in terms of the mean DGD

τ

as follows:

$\begin{matrix}{\left\langle {\left( {{\overset{\rightarrow}{\tau}}_{\omega} \cdot \hat{\tau}} \right)^{2}❘\tau} \right\rangle = {\frac{1}{27}{\left\langle \tau^{2} \right\rangle^{2}.}}} & (23)\end{matrix}$

Equation (23) can also be used to provide an independent measure of themean square DGD

τ²

by recognizing that the parallel component of the second order PMD inequation (23) is, equivalently, the frequency derivative of the DGD, asfollows:

$\begin{matrix}{{{\overset{\rightarrow}{\tau}}_{\omega} \cdot \hat{\tau}} = {\frac{\mathbb{d}{\overset{\rightarrow}{\tau}}}{\mathbb{d}\omega}.}} & (24)\end{matrix}$

The frequency derivative may be calculated from the scalar DGD vs.frequency measurements performed by a standard non-vectorial instrument,allowing the technique to be used with conventional PMD test sets. Asbefore, to avoid systematic errors, equation (23) may be combined withequation (24) and rewritten after application of the PDF of the scalarSOPMD as:

$\begin{matrix}{{{B_{1}\left\langle {\frac{\mathbb{d}{\overset{\rightarrow}{\tau}}}{\mathbb{d}\omega}} \right\rangle} = \tau_{RMS}^{2}},{and}} & (25) \\{{B_{2}\left\langle {\frac{\mathbb{d}{\overset{\rightarrow}{\tau}}}{\mathbb{d}\omega}}^{\frac{1}{2}} \right\rangle} = {\left\langle \tau \right\rangle.}} & (26)\end{matrix}$These relationships provide another unbiased evaluation of the mean DGDusing the frequency derivative of the DGD, and can be verified withsimulations. The coefficients

$B_{1} = {{\frac{3\pi}{2\log\; 2} \approx {6.80\mspace{14mu}{and}\mspace{14mu} B_{2}}} = {\left\lbrack {\frac{1}{2}{\int_{0}^{\infty}{\sqrt{y}\sec\; h^{2}y\ {\mathbb{d}y}}}} \right\rbrack^{- 1} \approx 2.64}}$do not depend on the fiber parameters, and may be calculated from thePDF derived for SOPMD, as described in Foschini et al.

While equations (20) and (21) require the SOPMD in vector form, found inaccordance with equation (18) above, equations (25) and (26) requireonly the scalar DGD, measured at discrete frequency intervals, so thatthe scalar SOPMD may be calculated, for example, in accordance withequation (19). Non-vectorial instruments that provide discrete ratherthan frequency-averaged scalar DGD measurements may, therefore, be usedto calculate τ² _(RMS) and

τ

from equations (25) and (26) respectively.

The improved accuracy of the method in accordance with the presentinvention for estimating the mean DGD

τ

and the mean square DGD

τ²

from the measured SOPMD, to that of the conventional averagingtechniques, has been demonstrated in simulations.

To estimate the measurement accuracy numerically, a simulation may beperformed on 10000 fibers consisting of 1000 random sections ofbirefringence. The polarization components, for example, Stokescomponents as well-known in the art, of the birefringence vector of eachsection in the simulation preferably have independent identical Gaussiandistribution. The delay of all birefringent sections is preferablynormalized to ensure that the true mean DGD is equal to τ_(RMS), and toprevent additional variations in the measurement due to the differencein “true” mean DGD among fibers. The PMD vector and its frequencyderivative are preferably computed at 256 angular frequency pointsseparated by

${{\Delta\omega} = \frac{2\pi}{10\tau_{RMS}}},$which is much less than the DGD correlation bandwidth. This correspondsto the DGD-angular bandwidth product Bτ_(RMS)≈150. By averaging DGD andthe other quantities in equations (20), (21), (25), and (26) over adifferent number of frequency points for every fiber, the effect thatthe bandwidth of the optical source has on the accuracy of the PMDmeasurement can be evaluated. It is expected that all simulatedmeasurements will be centered at

τ

independently of the measurement bandwidth used.

FIG. 5 shows a plot of the normalized standard deviations of the meanDGD measurements obtained by direct averaging of DGD (solid line) 70, byestimating

τ

from the vector SOPMD using equation (21) (dashed line) 72, and from thescalar SOPMD using equation (26) (dotted line) 74. As shown in FIG. 5,the mean DGD estimated from the second order PMD, whether estimated fromthe vector or scalar SOPMD, provides significantly better accuracy overthe prior art method of direct averaging.

Referring to FIGS. 6 a and 6 b, reductions in the standard deviationachieved by applying the statistical SOPMD to estimates of the mean DGDare shown. In FIGS. 6 a and 6 b, the number of samples 77 (pps) isplotted as a function of the mean DGD calculated in FIG. 6 b from thevector SOPMD 75, and as a function of the mean DGD calculated in FIG. 6a directly as an average 79. The standard deviation shown in FIG. 6 afor the averaged DGD 79 is about 0.044 ps, as compared with a standarddeviation shown in FIG. 6 b of 0.028 pps for the mean DGD derived fromSOPMD. This amounts to a factor of about a √{square root over (2)}reduction in the standard deviation, which corresponds to a two-foldreduction of the measurement variance. This reduction is comparable tothe effect that would be achieved by doubling the normalized measurementbandwidth Bτ_(RMS) in standard PMD measurement techniques.

FIG. 7 shows a plot of the normalized standard deviations for the meansquare DGD

τ²

_(B), measured directly (solid line) 80, and by estimating the meansquare DGD from the vector SOPMD using equation (20) (dashed line) 82,and from the scalar SOPMD using equation (25) (dotted line) 84. Asignificant reduction of the measurement uncertainty of at least 30% isshown in the simulation plotted in FIG. 7.

These numerical simulations also show that the mean DGD measurementstaken by different methods are not entirely correlated. Averaging theresults of these estimation techniques, therefore, allows furtherreduction of the measurement uncertainty. FIG. 8 presents theincremental improvements in the measurement accuracy obtained bycombining measurement techniques. Shown in dashed curve 90 is thestandard deviation of the measurements of A₂

|{right arrow over (τ)}_(ω)|^(1/2)

=

τ

, which represents the preferred method of calculating

τ

of those shown in FIG. 5. The dotted curve 92 represents the standarddeviation resulting from combining the estimates of DGD

τ

from the vector SOPMD 72 and the scalar SOPMD 74 (see FIG. 5). Marginalimprovement in the standard deviation is shown by combining techniquesas shown by dotted curve 92. The solid curve 94 additionally averages inthe result from direct averaging of the DGD 70 (see FIG. 5), which alsoshows some improvement over the use of the single technique 90.

An alternate method of the present invention includes combining thestatistical results of more than one of the methods provided herein tofurther reduce the error in measuring the DGD. Specifically, a linearcombination for the mean differential group delay DGD

τ

is preferably obtained from the LHS of equations (21) and (26), toobtain a measurement of mean differential group delay DGD

τ

. Coefficients of each term in the linear combination are preferablynormalized so that a sum of the coefficients is substantially equal toone. Similarly, a linear combination for the mean square differentialgroup delay DGD

τ²

_(B) is preferably obtained from the LHS of equations (20) and (25), toobtain a measurement of mean square differential group delay DGD

τ²

_(B). Coefficients of each term in the linear combination are preferablynormalized so that a sum of the coefficients is substantially equal toone. Analyses of the statistics of even higher orders of the PMD mayalso offer improvement over the prior art.

The methods provided herein in accordance with the present invention maybe applied to more accurately measure the true mean DGD in a fiber linkor a whole fiber route in an optical telecommunication system.

Although illustrative embodiments of the present invention have beendescribed herein with reference to the accompanying drawings, showingresults of applying the statistics up to the second-order PMD to moreaccurately measure the differential group delay of fiber, it is to beunderstood that the invention is not limited to those preciseembodiments, and that various other changes and modifications may beeffected therein by one skilled in the art without departing from thescope or spirit of the invention.

1. A method of determining a mean square differential group delay τ² _(RMS) of at least one length of optical fiber, the method comprising: measuring a polarization mode dispersion vector as a function of frequency, using a frequency-domain polarization mode dispersion measurement apparatus; calculating a second-order polarization mode dispersion vector {right arrow over (τ)}_(ω) as a function of frequency by calculating a derivative of the polarization mode dispersion vector with respect to frequency ω; calculating the mean of the magnitude of the second-order polarization mode dispersion vector |{right arrow over (τ)}_(ω)| to obtain a first result, according to

|{right arrow over (τ)}_(ω)|

; multiplying a proportionality coefficient A₁ by the first result to calculate the mean square differential group delay, in accordance with the following equation: A₁

|{right arrow over (τ)}_(ω)|

=τ² _(RMS),   (20) where A₁ is dimensionless, |{right arrow over (τ)}_(ω)| is in units of second² and τ² _(RMS) is in units of second²; and outputting a value of said mean square differential group delay.
 2. The method of claim 1, wherein A₁ is substantially equal to 2.02.
 3. The method of claim 1, wherein the frequency-domain polarization mode dispersion measurement apparatus comprises one of a polarimetric device and a Fixed Analyzer device.
 4. The method of claim 1, wherein the at least one length of optical fiber is a single optical fiber link.
 5. The method of claim 1, wherein the at least one length of optical fiber is an optical fiber route.
 6. A method of determining a mean square differential group delay τ² _(RMS) of at least one length of optical fiber, the method comprising: measuring a magnitude of a polarization mode dispersion vector as a function of frequency using a frequency-domain polarization mode dispersion measurement apparatus, the magnitude of the polarization mode dispersion vector being a scalar differential group delay; calculating a frequency derivative of the scalar differential group delay from the magnitude of the polarization mode dispersion vector, the frequency derivative of the scalar differential group delay $\frac{\mathbb{d}{\overset{\rightarrow}{\tau}}}{\mathbb{d}\omega}$  being a scalar second-order polarization mode dispersion function; calculating a first result, according to $\left\langle {\frac{\mathbb{d}{\overset{\rightarrow}{\tau}}}{\mathbb{d}\omega}} \right\rangle;$ multiplying a proportionality coefficient B₁ by the first result to calculate the mean square differential group delay, in accordance with the following equation: $\begin{matrix} {{{B_{1}\left\langle {\frac{\mathbb{d}{\overset{\rightarrow}{\tau}}}{\mathbb{d}\omega}} \right\rangle} = \tau_{RMS}^{2}},} & (25) \end{matrix}$ where B₁ is dimensionless, and $\frac{\mathbb{d}{\overset{\rightarrow}{\tau}}}{\mathbb{d}\omega}$ is in units of second²; and outputting a value of said mean square differential group delay.
 7. The method of claim 6, wherein B₁ is substantially equal to 6.80.
 8. The method of claim 6, wherein the frequency-domain polarization mode dispersion measurement apparatus comprises one of a polarimetric device and a Fixed Analyzer device.
 9. The method of claim 6, wherein the at least one length of optical fiber is a single optical fiber link.
 10. The method of claim 6, wherein the at least one length of optical fiber is an optical fiber route.
 11. A method of determining a mean square differential group delay τ² _(RMS) corresponding to a length of optical fiber, the method comprising: deriving a first mean square differential group delay τ² _(RMS) by: measuring a polarization mode dispersion vector as a function of frequency, using a frequency-domain polarization mode dispersion measurement apparatus; calculating a second-order polarization mode dispersion vector {right arrow over (τ)}_(ω) as a function of frequency by calculating a derivative of the polarization mode dispersion vector with respect to frequency ω; calculating the mean of the magnitude of the second-order polarization mode dispersion vector |{right arrow over (τ)}_(ω) to obtain a first result, according to

|{right arrow over (τ)}_(ω)|

; and multiplying a proportionality coefficient A₁ by the first result to calculate the mean square differential group delay, in accordance with the following equation: A₁

|{right arrow over (τ)}_(ω)|

=τ² _(RMS),   (20) where A₁ is dimensionless, |{right arrow over (τ)}_(ω)| is in units of second² and τ² _(RMS) is in units of second²; deriving a second mean square differential group delay τ² _(RMS) by: measuring a magnitude of a polarization mode dispersion vector as a function of frequency using a frequency-domain polarization mode dispersion measurement apparatus, the magnitude of the polarization mode dispersion vector being a scalar differential group delay; calculating a frequency derivative of the scalar differential group delay from the magnitude of the polarization mode dispersion vector, the frequency derivative of the scalar differential group delay $\frac{\mathbb{d}{\overset{\rightarrow}{\tau}}}{\mathbb{d}\omega}$  being a scalar second-order polarization mode dispersion function; calculating a first result, according to $\left\langle {\frac{\mathbb{d}{\overset{\rightarrow}{\tau}}}{\mathbb{d}\omega}} \right\rangle;$  and multiplying a proportionality coefficient B₁ by the first result to calculate the mean square differential group delay, in accordance with the following equation: $\begin{matrix} {{{B_{1}\left\langle {\frac{\mathbb{d}{\overset{\rightarrow}{\tau}}}{\mathbb{d}\omega}} \right\rangle} = \tau_{RMS}^{2}},} & (25) \end{matrix}$ where B₁ is dimensionless, and $\frac{\mathbb{d}{\overset{\rightarrow}{\tau}}}{\mathbb{d}\omega}$ is in units of second²; deriving a linear equation of the first mean square differential group delay τ² _(RMS) and the second mean square differential group delay τ² _(RMS) to calculate a combined mean square differential group delay τ² _(RMS), wherein a sum of coefficients of the linear equation is substantially equal to one; and outputting a value of said mean square differential group delay. 